Why does PCA of the vertices of a hexagon result in principal components of equal length?
Assuming that the $6$ vertices of the hexagon are on the unit circle,
&gt;&gt;&gt; from sympy import *
&gt;&gt;&gt; A = Matrix([[ 1, Rational(1,2),-Rational(1,2), -1, -Rational(1,2), Rational(1,2)], 
                [ 0,     sqrt(3)/2,     sqrt(3)/2,  0,     -sqrt(3)/2,    -sqrt(3)/2]])
&gt;&gt;&gt; A * A.T
Matrix([[3, 0],
        [0, 3]])

Since ${\rm A} {\rm A}^\top - 3 \, {\rm I}_2 = {\rm O}_2$, any two orthogonal directions could be the principal components.